Explicit formulation for the Dirichlet problem for parabolic-hyperbolic conservation laws

Boris Andreianov; Mohamed Karimou  Gazibo

doi:10.3934/nhm.2016.11.203

Article Contents

2016, Volume 11, Issue 2: 203-222. Doi: 10.3934/nhm.2016.11.203

This issue Previous Article Special issue on contemporary topics in conservation laws Next Article Relaxation approximation of Friedrichs' systems under convex constraints

Explicit formulation for the Dirichlet problem for parabolic-hyperbolic conservation laws

Boris Andreianov^1, and
Mohamed Karimou Gazibo^2,

1.
Laboratoire de Mathématiques de Besançon, Université de Franche-Comté, 25030 Besançon Cedex
2.
Université Abdou Moumouni de Niamey, École Normale Supérieure, Departement de Mathématiques, BP: 10963 Niamey

Received: May 2015

Revised: October 2015

Published: June 2016

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Abstract

We revisit the Cauchy-Dirichlet problem for degenerate parabolic scalar conservation laws. We suggest a new notion of strong entropy solution. It gives a straightforward explicit characterization of the boundary values of the solution and of the flux, and leads to a concise and natural uniqueness proof, compared to the one of the fundamental work [J. Carrillo, Arch. Ration. Mech. Anal., 1999]. Moreover, general dissipative boundary conditions can be studied in the same framework. The definition makes sense under the specific weak trace-regularity assumption. Despite the lack of evidence that generic solutions are trace-regular (especially in space dimension larger than one), the strong entropy formulation may be useful for modeling and numerical purposes.

Keywords:

Degenerate parabolic-hyperbolic equation,
Dirichlet boundary condition,
strong entropy formulation,
weakly trace-regular solutions,
uniqueness proof.

Mathematics Subject Classification: Primary: 35M13; Secondary: 35L04.

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